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Characterization of the essential spectrum of the
Neumann-Poincaré operator in 2D domains with
corner via Weyl sequences
Eric Bonnetier ∗ Hai Zhang, †
March 15, 2018
Abstract
The Neumann-Poincaré (NP) operator naturally appears in
thecontext of metamaterials as it may be used to represent the
solutionsof elliptic transmission problems via potentiel theory. In
particular,its spectral properties are closely related to the
well-posedness of thesePDE’s, in the typical case where one
considers a bounded inclusionof homogeneous plasmonic metamaterial
embedded in a homogeneousbackground dielectric medium. In a recent
work [30], M. Perfekt andM. Putinar have shown that the NP operator
of a 2D curvilinear poly-gon has an essential spectrum, which
depends only on the angles ofthe corners. Their proof is based on
quasi-conformal mappings andtechniques from complex-analysis. In
this work, we characterize thespectrum of the NP operator for a 2D
domain with corners in terms ofelliptic corner singularity
functions, which gives insight on the behaviorof generalized
eigenmodes.
1 Introduction
Plasmonic metamaterials are composite structures, in which some
parts aremade of media with negative indices. Their fascinating
properties of sub-wavelength confinement and enhancement of
electro-magnetic waves have
∗Laboratoire Jean Kuntzmann, Université Grenoble-Alpes, 700
Avenue Centrale, 38401Domaine Universitaire de
Saint-Martin-d’Hères, France, ([email protected]).†Department
of Mathematics, Hong Kong University of Science and Technology,
Clear
Water Bay, Kowloon, HK, ([email protected]).
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drawned considerable interest from the physics and mathematics
communi-ties. The progress in the controled production of
composites with character-istic features of the order of optical
wavelengths contributes to this activity,as it may enable many
applications to nano-optical-mechanical systems, can-cer therapy,
neuro-science, energy and information storage and processsing.
From the mathematical modeling point of view, these studies have
also re-newed interest in the Neumann-Poincaré operator, the
integral operatorderived from the normal derivative of the single
layer potential. Indeed, itproves to be an interesting tool to
construct, represent and derive propertiesof solutions to
diffusion-like equations, in situations where the Lax-Milgramtheory
does not apply, which is typically the case of negative index
materials.
The spectral properties of this operator have proved interesting
in severalcontexts [1, 2, 12, 13, 14]. They are particularly
relevant to metamaterials,as they are closely related to the
existence of surface plasmons, i.e., solutionsof the governing PDE
(Maxwell, Helmholtz, accoustic equations) which aresupported in the
vicinity of the interfaces where the coefficients change signs.
To fix ideas, we consider a single inclusion D made of negative
index ma-terial (typically metals, such as gold or silver at
optical frequencies). Itis embedded in a homogeneous dielectric
background medium and we bydenote K∗D the associated NP operator
(its precise definition is given in sec-tion 2). For particular
frequencies, called plasmonic resonant frequencies, anincident wave
may excite electrons on the surface of the inclusion into a
res-onant state, that generates highly oscillating and localized
electromagneticfields. For gold and silver, plasmonic resonances
occur when the diameter ofthe particles is small compared to the
wavelength. From the modeling pointof view, one may rescale the
governing Maxwell or Helmholtz equations,with respect to particule
size, and take the limit of the resulting equationsto obtain the
quasi-static regime, where only the higher-order terms of
theoriginal PDE remain [27, 19, 3, 4]. Plasmonic resonances have
been investi-gated via layer potential techniques in [1]–[6].
When D has a smooth boundary (say C2) the operator K∗D is
compact.Its spectrum is real, contained in the interval (−1/2,
1/2], and consists ina countable number of eigenvalues that
accumulates to 0. In the contextof plasmonics, domains with corners
present an obvious interest when oneattemps to concentrate
electro-magnetic fields, and several authors have
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considered geometries where the negative index materials are
distributedin regions with corners [10, 11, 21, 22]. When D has
corners, K∗D is notcompact [31]. In a recent work, M.-K. Perfekt
and M. Putinar have shown,relying on the relationship betwen
complex analysis and potential theory,that the NP operator
associated to a planar domain with corners has essen-tial spectrum,
which they characterized to be
σess(K∗D) = [λ−, λ+], λ+ = −λ− =1
2(1− α
π),
where α is the most acute angle of D. See [30, 29].
The objective of our paper is to give an alternative derivation
of the essentialspectrum of K∗D when D has corners, and to
establish a close connectionbetween the fact that K∗D has essential
spectrum and the theory of ellipticcorner singularities initiated
by Kondratiev in the 1970’s and developed inmany directions. See
[25] and also [20, 16, 26] and the many referencestherein. This
theory shows that the solution u to an elliptic scalar equationin a
domain O with corners splits as the sum u = ureg+using of a regular
partureg ∈ H2(O) and a singular part using ∈ H1(O) \ H2(O), locally
aroundeach corner. Up to a scaling factor, the expression of the
latter part, whichwe call ‘singularity function’, only depends on
the geometry of the corner,and on the nature of the boundary
conditions. In the case of a transmissionproblem, it depends on the
angle and on the contrast in material coefficients.Typically, using
is a non-trivial solution of a homogeneous problem for
theassociated operator in the infinite domain obtained by zooming
around thevertex of the corner. For a transmission problem in 2D,
it has the form
using = Crηϕ(θ), (1)
where (r, θ) denote the polar coordinates with orgin at the
vertex of thecorner under consideration. The exponent η is the root
of a dispersionrelation, and ϕ is a smooth function (or piecewise
smooth in the case of atransmission problem).
This paper is organized in the following way. Section 2 of the
paper describesthe setting and notations and reviews useful facts
about the NP operator. InSection 3, we study how elliptic corner
singularity functions depend on theconductivity contrast. In the
very interesting papers [9, 10, 11], it is shownthat functions of
the form (17) only exist when the conductivity contrast λlies
outside a critical interval [λ−, λ+]. When λ ∈ [λ−, λ+], the
elliptic cornersingularity functions still have the form using but
their expression involves
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a complex exponent η. In [11], the use of the Mellin transform
convertsthe search of these singular functions to that of
propagative mode in aninfinite wave-guide. These functions are
called plasmonic black-hole waves,reflecting the fact that they are
not in the energy space H1(Ω). In Section 4,we show that the
critical interval is contained in the essential spectrumσess(K∗D),
by generating singular Weyl sequences [8] using the
singularityfunctions. In Section 5, the reverse inclusion is
proved. In particular, we usea construction inspired by [28] to
transform, around the vertex of the corner,the PDE with sign
changing conditions into a system of PDE’s defined inthe
inhomogeneity only, that satisfies complementing boundary
conditionsin the sense of Agmon, Douglis and Nirenberg, and for
which we provewell-posedness.
2 The Neumann-Poincaré operator and the Poin-caré variational
operator
Throughout the text, Ω ⊂ R2 denotes a bounded open set with
smoothboundary, that strictly contains a connected inclusion D. We
assume that∂D is smooth, except for one corner point, of angle α, 0
< α < Pi, locatedat the origin. We assume that for some R0
> 0,
D ∩BR0 = {x = (r cos(θ), r sin(θ)), 0 ≤ r < R0, |θ| <
α/2}, (2)
where, for any ρ > 0, Bρ denotes the ball of radius ρ
centered at 0. Thespace H10 (Ω) is equipped with the following
inner product and associatednorm
< u, v >H10 =
∫Ω∇u · ∇v dx, ||u||H10 =
(∫Ω|∇u|2 dx
)1/2.
Our work concerns the following diffusion equation: given a
function f ∈ L2(Ω),we seek u such that{
−div(a(x)∇u(x)) = f in Ω,u(x) = 0 on ∂Ω,
(3)
where the conductivity a is piecewise constant
a(x) =
{k ∈ C x ∈ D,
1 x ∈ Ω \D. (4)
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It is well known that when k is strictly positive, or when k ∈ C
andIm(k) 6= 0, this problem has a unique solution in H1(Ω), and
that
||u||H10 ≤ C(k) ||f ||L2 ,
for some constant C(k) > 0 that depends on k.
Let P (x, y) denote the Poisson kernel associated to Ω, defined
by
P (x, y) = G(x, y) +Rx(y), x, y ∈ Ω,
where G(x, y) denotes the free space Green function
G(x, y) =1
2πln |x− y|,
and where Rx(y) is the smooth solution to{∆yRx(y) = 0 y ∈
Ω,Rx(y) = −G(x, y) y ∈ ∂Ω.
With the Poisson kernel, we define the single layer potentials
SDϕ ∈ L2(∂D)of a function ϕ ∈ L2(∂D) by
SDϕ(x) =∫∂D
P (x, y)ϕ(y) ds(y), x ∈ D ∪ (Ω \D).
It is well known [18, 31] that SDϕ is harmonic in D and in Ω \D,
continousin Ω, and that its normal derivatives satisfy the Plemelj
jump conditions
∂SDϕ∂ν|±(x) = (±1
2I +K∗D)ϕ(x), x ∈ ∂D. (5)
where K∗D is the Neumann-Poincaré operator, defined by
K∗Dϕ(x) =∫∂D
∂P
∂νy(x, y)ϕ(y) ds(y).
It is shown in [15] that this definition makes sense for
Lipschitz domains,and in that case, the operator K∗D is continuous
from L2(∂D) → L2(∂D),which extends as an operator H−1/2(∂D)→
H1/2(∂D).
The solution u to (3) can then be represented in the form
u(x) = SDϕ(x) +H(x), (6)
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where the harmonic part is given by
H(x) =
∫ΩP (x, y)f(y) ds(y).
The jump conditions (5), constrain the layer potential ϕ ∈
H−1/2(∂D) tosatisfy the integral equation
(λI −K∗D)ϕ(x) = ∂νH|∂D(x), x ∈ ∂D.
We also introduce the Poincaré variational operator TD : H10
(Ω)→ H10 (Ω),
defined for u ∈ H10 (Ω) by
∀ v ∈ H10 (Ω),∫
Ω∇TDu · ∇v dx =
∫D∇u · ∇v dx. (7)
Some of its properties are described in the following
proposition (see [12] fora proof).
Proposition 1. The operator TD is bounded, selfadjoint, and
satisfies ||TD|| = 1.Moreover,
(i) Its spectrum σ(TD) is contained in the interval [0, 1].
(ii) Its kernel, the eigenspace associated to β = 0, is
Ker(TD) = {u ∈ H10 (Ω), u = const on D}.
(iii) 1 ∈ σ(TD) and the associated eigenspace is
Ker(I − TD) = {u ∈ H10 (Ω), u = 0 in Ω \D},
(and thus, can be identified with H10 (D)).
(iv) The space H10 (Ω) decomposes as
H10 (Ω) = Ker(TD)⊕Ker(I − TD)⊕H,
where H is the closed subspace defined by
H = {u ∈ H10 (Ω),∆u = 0 in D ∪ (Ω \D),∫∂D
∂u+
∂νds = 0}.
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The space Hs = H ⊕Ker(TD) is the space of single layer
potentials. It isisomorphic to
H−1/20 (∂D) = {ϕ ∈ H
1/2(∂D), < ϕ, 1 >H−1/2,H1/2= 0}.
This latter space is equipped with the inner product
< ϕ,ψ >S = −∫∂D
ϕSψ dσ (8)
for which the operator K∗D : H−1/20 (∂D) → H
−1/20 (∂D) is self-ajdjoint as
a result of the Calderón identity [24]. We denote by || · ||S
the associatednorm. In particular, if u, v ∈ Hs are such that u =
SDϕ, v = SDψ, then thejump conditions (5) and integration by parts
show that∫
Ω∇u · ∇v = < ϕ,ψ >S . (9)
When the domain D has a C2 boundary, the Poincaré-Neumann
operatorK∗D : H
−1/20 (∂D) → H
−1/20 (∂D) is compact. Its spectrum σ(K∗D) is con-
tained in [−1/2, 1/2], and consists of a sequence of real
eigenvalues thataccumulates to 0. In this case, σ(K∗D) is directly
related to σ(TD). Indeed,if u ∈ H10 (Ω) and β ∈ R, β 6= 1, satisfy
TDu = βu, it follows from (7) that
∀ v ∈ H10 (Ω), β∫
Ω\D∇u · ∇v dx+ (β − 1)
∫D∇u · ∇v dx = 0,
so that u is a non-zero solution to{div(a(x)∇u(x)) = 0 in Ω,
u(x) = 0 on ∂Ω,(10)
where the conductivity a equals β in Ω \D and (β − 1) in D.
Expressing uin the form u = SDϕ yields yields the integral
equation
(λI −K∗D)ϕ(x) = 0, x ∈ ∂D,
where λ = 1/2− β is thus an eigenvalue of K∗D. It follows
that
σ(TD) = (1/2− σ(K∗D)) ∪ {0, 1}
As recalled above, when D is a domain with corners, σ(K∗D)
contains aninterval of essential spectrum [30]. We have
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Proposition 2. The essential spectra of TD and K∗D are related
by σess(TD) =1/2− σess(K∗D).
Proof: Let λ ∈ σess(K∗D). By definition, there exists a singular
Weyl se-quence, i.e., a sequence of functions (ϕε) ⊂ H−1/20 such
that
(λI −K∗D)ϕε → 0 strongly in H−1/20 ,
||ϕε||S = 1,ϕε ⇀ 0 weakly in H
−1/20 .
Let β = 1/2 − λ and uε = SDϕε ∈ HS . Let v ∈ HS so that v = SDψ
forsome ψ ∈ H−1/20 (∂D). It follows from (9) that∫
Ω∇uε · ∇v = < ϕε, ψ >→ 0.
This equality also holds for v ∈ Ker(I−TD) since this subspace
is orthogonalto HS , and thus
uε ⇀ 0 weakly in H10 (Ω). (11)
Additionally, invoking (9) again, we see that∫Ω|∇uε|2 = < ϕε,
ϕε >S = 1. (12)
Finally, we compute, for v = SDψ ∈ HS ,∫Ω∇ ((βI − TD)uε) · ∇v
=
∫Ωβ∇uε · ∇v −
∫D∇uε · ∇v
=
∫Ω\D
β∇uε · ∇v +∫D
(β − 1)∇uε · ∇v
= −β∫∂D
∂νuε|+v + (β − 1)∫∂D
∂νuε|−v.
Inserting (5) in place of the normal derivatives of uε we see
that∫Ω∇ ((βI − TD)uε) · ∇v = < (λI −K∗D)ϕε, ψ >S
≤ ||(λI −K∗D)ϕε||S ||ψ||S .
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It follows that
||(βI − TD)uε||H1 ≤ ||(λI −K∗D)ϕε||S → 0. (13)
we conclude from (11–13) that uε is a singular Weyl sequence
associated toβ, so that β ∈ σess(TD). The same argument proves the
reverse inclusionσess(TD) ⊂ (1/2− σess(K∗D)).
3 Corner singularity functions
Elliptic corner singularities have been the subject of much
research sincethe pionneering works of Kondratiev [25], Grisvard
[20] (see also [16, 26]).Essentially, the theory focuses on the
regularity of solutions to elliptic PDEsnear a corner of the
domain, or in the case of a transmission problem suchas (3), near a
corner of the interface between several phases. The followingis a
typical statement:
Theorem 1. Let k > 0. The solution u ∈ H10 (Ω) to (3)
decomposes as
u = using + ureg,
where ureg ∈ H2(Ω) and where using has the form
using(x) = rηϕ(θ)ζ(x), x ∈ Ω. (14)
Here x = (r cos(θ), r sin(θ)) in polar coordinates, ζ is a
smooth cut-off func-tion, such that, for some s > 0
ζ(x) =
{1 |x| < s,0 |x| > 2s.
Moreover, for some constant C = C(α, k), the following estimate
holds
||using||H1(Ω) + ||ureg||H2(Ω) ≤ C(||u||H1(Ω) + ||f ||L2(Ω)
). (15)
These results have been derived only in the case of elliptic
media, i.e., whenk > 0. In the context of plasmonic
metamaterials, it is natural to try toextend them to complex values
of k. To our best knowledge, the first stepsin this direction have
been obtained in [9, 10] and concern the existence ofsingularity
functions of the form (14).
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3.1 Regular corner singularity functions
In this paragraph, we investigate whether one can define
singular functionssuch as (14) when k may also take negative
values. More precisely, we seekH1loc(R
2) solutions to
div(a(x)∇u(x)) = 0 in R2, (16)
of the form
u(x) = rηϕ(θ), η ∈ R, (17)
when the conductivity a(x) is defined in the whole of R2 by
a(x) =
{k |θ| < α/2,1 otherwise.
(18)
Since we are only interested in singular solutions which belong
to H1(Ω) \H2(Ω), we may restrict η to lie in (0, 1). As u is
harmonic in each sector|θ| < α/2 and α/2 < θ < 2π − α/2,
it follows that ϕ has the form
ϕ =
{a1 cos(η(θ + α/2)) + b1 sin(η(θ + α/2)) if − α/2 < θ <
α/2,a2 cos(η(θ + α/2)) + b2 sin(η(θ + α/2)) if α/2 < θ < 2π −
α/2
(19)
for some ai, bi, i = 1, 2. Expressing the continuity of u and of
a(x)∂νuacross the interfaces, shows that a non-trivial solution
exits if and only ifthe following dispersion relation is
satisfied
det
1 0 − cos(2πη) − sin(2πη)
cos(αη) sin(αη) − cos(αη) − sin(αη)0 k sin(2πη) − cos(2πη)
−k sin(αη) k cos(αη) − sin(αη) − cos(αη)
= 0,which, after elementary manipulations, can be rewritten in
the form
2k
k2 + 1=
sin(αη) sin((2π − α)η)1− cos(αη) cos((2π − α)η)
=: F (η, α). (20)
A Taylor expansion around the values η = 0 shows that F (·, α)
can beextended by continuity to a function defined on the whole of
[0, 1] by setting
F (0, α) =−2α(2π − α)α2 + (2π − α)2
.
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By solving2k
k2 + 1= F (0, α),
we obtain two solutions
k+ =−(2π − α)
α, k− =
−α2π − α
. (21)
Additionnally, it is easy to check that |F (η, α)| ≤ 1 and
∂ηF =cos((2π − α)η)− cos(αη)) [a sin((2π − α)η)− (2π − α)
sin(αη)]
[1− cos(αη) cos((2π − α)η)]2.
We show below that F (·, α) is thus strictly increasing, and
note that ∂ηF (0, α) =∂ηF (1, α) = 0.
Lemma 1. For any 0 < α < π and 0 ≤ η ≤ 1, the following
inequalitieshold
cos((2π − α)η)− cos(αη) < 0, (22)a sin((2π − α)η)− (2π − α)
sin(αη) < 0.
Proof: To prove the first inequality, we first note that α <
(2π−α) so thatαη < (2π − α)η. If (2π − α)η ≤ π, then (22)
follows from the monotonicityof the cosine function on [0, π]. If
(2π − α)η > π, then
cos((2π − α)η) = cos(π − β), with (2π − α)η =: π + β.
Noticing that
αη ≤ αη + 2π(1− η) = π − β < π,
we infer that cos(π − β) < cos(αη), which yields the
result.
The second inequality follows from the fact that
∂η [a sin((2π − α)η)− (2π − α) sin(αη)]= α(2π − α) [cos((2π −
α)η)− cos(αη)] ,
which according to (22) is negative.
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As a consequence of (20), we obtain
Proposition 3. Singular solutions in H1loc(R2) of the form (17)
exists for
the equation (16)only when k ∈ (−∞, k+) ∪ (k−,+∞), see Figure
3.1. Interms of the contrast λ = k+12(k−1) this condition is
equivalent to
λ /∈ [λ−, λ+] := [−1
2(1− α
π),
1
2(1− α
π)].
In other words, singular solutions of the form (17) only exist
when λ =k+1
2(k−1) is not in σess(K∗D).
Figure 1: Left: Plot of the function k → 2k/(k2 + 1). Right:
Plot ofη → F (η, α) (blue), and of ξ → F̃ (ξ, α) (green), for α =
π/2. The dottedline indicates the value of 2k/(k2 + 1) below which
the dispersion relationhas no solution η ∈ R.
3.2 Singular corner singularity functions
We now construct local singular solutions when k ∈ [k+, k−]. By
this wemean functions which satisfy the PDE (16), but which may
only be inH1loc(R
2 \{0}). To this end, we seek u(x) = rηϕ(θ), with ϕ in the form
(19),but assume now that η ∈ C. The same algebra leads to the same
dispersionrelation (20). In particular if we restrict η to be a
pure imaginary number,η = iξ, this relation takes the form
2k
k2 + 1=
sinh(αξ) sinh((2π − α)ξ)1− cosh(αξ) cosh((2π − α)ξ)
=: F̃ (ξ, α).
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It is easy to check that the function ξ → F̃ (ξ, α) can be
extended by conti-nuity at ξ = 0 by setting
F̃ (0, α) =2α(2π − α)
α2 + (2π − α)2=
2k±k2± + 1
(and we note that F̃ (0, α) = F (0, α)). In addition, we
compute
∂ξF̃ =(cosh((2π − α)ξ)− cosh(αξ)) [(2π − α) sinh(αξ)− α sinh((2π
− α)ξ)]
[1− cosh(αξ) cosh((2π − α)ξ)]2.
Just as in Proposition 1, one can show that for any ξ > 0 and
0 < α < π,the product of the two factors in the above
numerator is negative, so thatF̃ (·, α) is strictly decreasing on
R+ and its range is equal to [F (0, α),−1),see Figure 3.1. We also
note that limη→∞ F̃ (η, α) = −1, so that the value−1 (which
corresponds to k = −1) is never attained. Summarizing, we haveshown
that
Proposition 4. For any value of λ ∈ (λ−, λ+), λ 6= 0, there
exists ξ > 0 anda function u(x) = riξϕ(θ), which is a local
solution to div(a(x)∇u(x)) = 0,where a is defined by (18), with λ =
k+12(k−1) .
4 Construction of singular Weyl sequences
In this section we prove
Theorem 2. The set [λ−, λ+] is contained in σess(TD).
Proof:Since σess(K∗D) is a closed set, it is sufficient to show
that (λ−, λ+) \ {0} ⊂σess(TD). We proceed as follows: We consider λ
∈ (λ−, λ+), λ 6= 0, and showthat β = 1/2− λ ∈ σess(TD) by
constructing a singular Weyl sequence, i.e.,a sequence of functions
uε ∈ H10 (Ω), such that
||uε||H10 = 1,(βI − TD)uε → 0 strongly in H10 (Ω),
uε ⇀ 0 weakly in H10 (Ω).
(23)
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According to Proposition 4, there exists ξ > 0 and
coefficients a1, b2, a2, b2 ∈C, not all equal to 0, such that the
function
u(x) = Re(riξ)φ(θ) =
Re(riξ) [a1 cos(iξ(θ + α/2)) + b1 sin(iξ(θ + α/2))]
if − α/2 < θ < α/2,
Re(riξ) [a2 cos(iξ(θ + α/2)) + b2 sin(iξ(θ +
α/2))]otherwise,
(24)
is harmonic in (D ∩ BR0) \ {0} and in((Ω \D) ∩BR0
)\ {0}, and satisfies
the transmission conditions at the interfaces θ = ±α/2.
Let r0 < R0/2 and let χ1, χ2 : R+ → [0, 1] denote two smooth
cut-off
functions, such that for some constant C > 0χ1(s) = 0 |s| ≤
1, χ2(s) = 0 |s| ≥ 2r0,χ1(s) = 1 |s| ≥ 2, χ2(s) = 1 |s| ≤
r0,|χ′1(s)| ≤ C, |χ′2(s)| ≤ C.
We set χε1(r) = χ1(r/ε), and define
uε(x) = sεχε1(r)χ2(r)u(x), x ∈ Ω. (25)
The function u is not in H1 as its gradient blows up like r−1
near the corner,consequently
mε :=
∫ r0ε
∫ 2π0|∇u(x)|2 rdrdθ → ∞ as ε→ 0.
We choose sε in (25) so that ||uε||H10 = 1, in other words
s−2ε =
∫ 2εε
∫ 2π0|u∇χε1 + χε1∇u|2 + mε +
∫ 2r0r0
∫ 2π0|u∇χ2 + χ2∇u|2
=: J1 +mε + J2.
The term J2 is independent of ε and is O(1), and in
particular
J2 = o(mε) as ε→ 0.
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The other term can be estimated as follows
J1 =
∫ 2εε
∫ 2π0
∣∣∣∣riξ + r−iξ2 ϕ(θ)χ′1(r/ε)/ε+ iξ riξ−1 − r−iξ−12
ϕ(θ)χ1(r/ε)∣∣∣∣2
+
∣∣∣∣riξ−1 − r−iξ−12 ϕ′(θ)χ1(r/ε)∣∣∣∣2 rdrdθ (26)
≤ C∫ 2π
0
(|ϕ(θ)|2 + |ϕ′(θ)|2
)dθ
∫ 2εε
(||χ′1||2∞/ε2 + r−2||χ1||∞
)rdr
≤ C∫ 2π
0
(|ϕ(θ)|2 + |ϕ′(θ)|2
)dθ(3/2||χ′1||2∞ + (ln(2ε)− ln(ε))||χ1||∞
).
Since ϕ is independent of ε, we see that
J1 = O(1) = o(mε), as ε→ 0,
and so sε ∼ m−1/2ε → 0.
We next show that ||(βI − TD)uε||H1 → 0. Indeed, let v ∈ H10 (Ω)
andconsider
J =
∫Ω∇(βI − TD)uε · ∇v
=
∫Ω\D
β∇uε · ∇v +∫D
(β − 1)∇uε · ∇v,
in view of the definition of TD. Inserting the expression (25)
of uε, we seethat
J = sε
∫Ω\D
β∇u · ∇(χε1χ2v) + sε∫D
(β − 1)∇u · ∇(χε1χ2v)
+ sε
∫Ω\D
βu∇(χε1χ2) · ∇v + sε∫D
(β − 1)u∇(χε1χ2) · ∇v
− sε∫
Ω\Dβ∇u · v∇(χε1χ2) − sε
∫D
(β − 1)∇u · v∇(χε1χ2).
Since u is a local solution to (16), the sum of the first 2
integrals vanishes,and we remain with
J =
(sε
∫Ωau∇(χε1χ2) · ∇v + sε
∫Ω∩(B2r0\Br0 )
au∇(χ2) · ∇v
)
+ sε
∫Ω∩(B2ε\Bε)
av∇(χε1) · ∇u =: sε(J3 + J4). (27)
15
-
where a = β in Ω \D and a = β − 1 in D. The Cauchy-Schwarz
inequalityallows us to estimate the first two terms on the
right-hand side by
|J3| ≤ Csε ||v||H1{∫ 2ε
0
∫ 2π0
(|u|2|χ′1|2/ε2 + |∇u|2|χ1|2
)rdrdθ
+
∫ 2r0r0
∫ 2π0
(|u|2|χ′2|2 + |∇u|2|χ2|2
)rdrdθ
}. (28)
and the same arguments as those used to control the term J1 in
(26) showthat the two integrals above are O(1). As for the last
term in (27), we write
J4 :=
∫B2ε\Bε
a∇u · v∇χε
=
∫B2ε\Bε
a∇u · v∇χε +∫B2ε\Bε
a∇u · (v − v)∇χε,
where v = |B2ε|−1∫B2ε
v(x) dx. We note that the first integral in the aboveright-hand
side reduces to
v
∫ 2π0
a(θ)φ(θ) dθ
∫ 2εε
iξ
(riξ−1 − r−iξ−1
2
)χ′1(r/ε)
εrdr = 0.
Indeed, since φ is a solution to (a(θ)φ′(θ))′ − ξ2a(θ)φ(θ) = 0,
with periodicboundary conditions, it satisfies∫ 2π
0a(θ)φ(θ) dθ = 0.
It follows that
|J4| ≤
(∫B2ε\Bε
a2|∇u · ∇χε|2 dx
)1/2(∫B2ε
|v − v|2)1/2
.
Using the following Poincaré inequality∫B2ε
|v − v|2 ≤ 4|B2ε|2∫B2ε
|∇v|2,
16
-
we obtain
|J4| ≤ Cε||v||H1(Ω)(∫ 2π
0a(θ)2|φ(θ)|2 dθ
)1/2(∫ 2ε
ε|iξ r
iξ−1 − r−iξ−1
2|2 [χ
′(r/ε)]2
ε2rdr
)1/2≤ C
(∫ 2εε
r−1dr
)1/2||v||H1(Ω)
≤ C√ln(2)||v||H1(Ω) = O(1)||v||H1(Ω).
Altogether, (27, 28) and the above estimate show that
∀ v ∈ H10 (Ω),∣∣∣∣ ∫
Ω∇(βI − TD)uε · ∇v
∣∣∣∣ ≤ O(sε)||v||H1 ,which proves the claim since sε →
0.Finally, we show that uε → 0 weakly in H1(Ω). In fact, since this
sequenceis uniformly bounded in H1, it suffices to show that uε → 0
strongly in L2,which follows from (25), from the boundedness of χ1
and χ2 and from thefact that sε → 0.
5 Characterization of the essential spectrum
In this section, we consider λ /∈ [λ−, λ+], β = 1/2 − λ, and k =
(1 − 1/β).The latter satisfies
k < k+ =−(2π − α)
α< 0 or k− =
−α2π − α
< k < 0. (29)
We show that β /∈ σess(TD), so that according to Proposition 2,
λ /∈ σess(K∗D).
We proceed by contradiction: If β ∈ σess(TD), then there exists
a singu-lar Weyl sequence uε, that satisfies the conditions (23).
In the next threesections, we show
Proposition 5. The sequence uε converges to 0 strongly in
H1(Ω).
This contradicts the fact that ||uε||H1 = 1. Consequently, in
view of Theo-rem 2, this proves
Theorem 3. The essential spectrum of K∗D is exactly
σess(K∗D) = [λ−, λ+].
17
-
5.1 Controling the energy of uε away from the corner
Let zε = βuε − TDuε ∈ H10 (Ω). Let ρ < R0 and let χρ denote a
smooth,radial cut-off function, such that
χρ(x) =
{1 if |x| ≤ ρ/2,0 if |x| ≥ ρ.
Let vε = (1− χρ)uε. We show that
Proposition 6. The sequence vε converges strongly to 0 in
H1.
Proof: Assume that it is not the case. Then there exists δ >
0 and asubsequence (still labeled with ε) such that
||vε||H10 ≥ δ. (30)
We note that for any v ∈ H10 (Ω),∫Ω∇zε · ∇v =
∫Ω∇(βuε − TDuε) · ∇v (31)
= β
∫Ω∇uε · ∇v +
∫D∇uε · ∇v
=
∫Ωa∇uε · ∇v, (32)
where a(x) = β, x ∈ Ω \D, and a(x) = β − 1, x ∈ D. Given v ∈ H10
(Ω), wecompute∫
Ωa∇vε · ∇v =
∫Ωa∇ [(1− χρ)uε] · ∇v
=
∫Ωa [(1− χρ)∇uε − uε∇χρ] · ∇v
=
∫Ωa∇uε · [∇ ((1− χρ)v) + v∇χρ] − auε∇χρ · ∇v
=
∫Ω∇zε · ∇ ((1− χρ)v) − uε∇ · (av∇χρ) − auε∇χρ · ∇v.
Invoking the Cauchy-Schwarz and the Poincaré inequality, it
follows that∣∣∣∣ ∫Ω∇ ((βI − TD)vε) · ∇v
∣∣∣∣ = ∣∣∣∣ ∫Ωa∇vε · ∇v
∣∣∣∣≤ C
(||uε||L2 + ||zε||H10
)||v||H10 ,
18
-
As uε → 0 strongly in L2(Ω) since it converges weakly to 0 in
H1, weconclude that
(βI − TD)vε = (1− χρ)uε → 0 strongly in H10 (Ω). (33)
We note that since vε has support in Ω \Bρ/2,
TDvε = TD̃vε,
where D̃ denotes any smooth connected inclusion, such that (D \
Bρ/2) ≡(D̃ \Bρ/2), and thus (33) also reads
(βI − TD̃)vε = (1− χρ)uε → 0 strongly in H10 (Ω).
It is easily seen that vε ⇀ 0 weakly in H10 (Ω), and, upon
rescaling in view
of (30), we conclude from the above estimate that vε/||vε||H10
is a singularWeyl sequence for TD̃. But D̃ is smooth, so that the
associated Neumann-Poincaré operator is compact and does not have
essential spectrum, whichcontradicts this fact, and proves the
Proposition.
5.2 Controling the energy of uε near the corner
We now focus on wε := χρuε, which has compact support in Bρ. In
viewof (31), it is easy to check that wε satisfies
∂2rrwε + 1/r∂rwε + 1/r2∂2θθwε = f̃ε,
in (D ∩Bρ) and in((Ω \D) ∩Bρ
). The right-hand side is defined as
f̃ε = χρ∆zε + b∇χρ · ∇uε +∇(buε) · ∇χρ + buε∆χρ,
and we note that it converges strongly to 0 in H−1(Ω). Moreover,
since thefunction χρ is radial, wε satisfies the following
transmission conditions onthe edges of the corner
wε(r,α2 |−) = wε(r,
α2 |+)
wε(r,−α2 |−) = wε(r,−α2 |+),
(β − 1)∂θwε(r, α2 |−) = β∂θwε(r,α2 |+),
(β − 1)∂θwε(r,−α2 |−) = β∂θwε(r,−α2 |+),
where the notations |−, |+ indicate taking the limit from left
and right sidesrespectively.
19
-
We set
A =α
2π − α∈ (0, 1), (34)
and consider the change of variables (r, θ) ∈ (0, ρ)× (−α/2,
α/2)→ (r, π −θ/A), which maps D ∩Bρ into (Ω \D) ∩Bρ. We define{
vε(r, θ) = wε(r, π − θ/A)g̃ε(r, θ) = f̃ε(r, π − θ/A),
for (r, θ) ∈ D ∩Bρ.
It is easy to check that when (f, g) = (f̃ε, g̃ε), the functions
(w, v) =(wε|D∩Bρ , vε) satisfy the following system{
∂2rrw + 1/r∂rw + 1/r2∂2θθw = f,
∂2rrv + 1/r∂rv +A2/r2∂2θθv = g,
(35)
with the boundary conditionsv(ρ, θ) = w(ρ, θ) = 0,v(r,±α/2) =
w(r,±α/2),∂θv(r,±α/2) = −kA ∂θw(r,±α/2).
(36)
In other words, wε and uε both satisfy an elliptic equation and
take nearlythe same Cauchy data on the edges of the corner.
To study the above system, we introduce the (closed) subspace V1
⊂ H1(D∩Bρ)×H1(D ∩Bρ) of functions (w, v) that satisfy{
v(ρ, θ) = w(ρ, θ) = 0 |θ| < α/2v(r,±α/2) = w(r,±α/2) 0 < r
< ρ.
Theorem 4. The system (35–36) has a unique solution (w, v) ∈ V1.
More-over, there exists a constant C > 0, such that
||∇w||L2(D∩Bρ) + ||∇v||L2(D∩Bρ) ≤ C(||f ||H−1(D∩Bρ) +
||g||H−1(D∩Bρ)
).
Proof: On V1 we consider the norm
||(w, v)|| :=
(∫D∩Bρ
|∇w|2 + |∇v|2)1/2
. (37)
20
-
We multiply the equations (35) by two functions φ, ψ ∈ H1(D ∩
Bρ) thatvanish on D ∩ ∂Bρ, and integrate to obtain∫
D∩Bρfφ + gψ
=
∫ ρ0
∫ α/2−α/2
(∂rw1r∂θw
)·(
∂rφ1r∂θφ
)+
(∂rvA2
r ∂θv
)·(
∂rψ1r∂θψ
)rdrdθ
−∫θ=±α/2
1
r∂θwφ +
A2
r∂θvψ.
We note that that the last integral can be rewritten
as∫θ=±α/2
1
r∂θw(φ−Akψ) +
A2
r
[∂θv +
k
A∂θw
]ψ.
To satisfy the natural boundary condition in (36), we are thus
led to intro-duce the subspace V2 ⊂ H1(D ∩Bρ)×H1(D ∩Bρ) of
functions (φ, ψ) thatsatisfy
φ(ρ, θ) = ψ(ρ, θ) = 0, |θ| < α/2
φ(r,±α2 )−Akψ(r,±α2 ) = 0, 0 < r < ρ,
which we also equip with the norm (37). We also introduce the
followingbilinear form B on V1 × V2 by
B
((wv
),
(φψ
))=
∫ ρ0
∫ α/2−α/2
(∂rw1r∂θw
)·(
∂rφ1r∂θφ
)+
(∂rvA2
r ∂θv
)·(
∂rψ1r∂θψ
)rdrdθ.
Thus, solving (35–36) amounts to solving the variational problem
: findW = (w, v) ∈ V1 such that
∀ Φ = (φ, ψ) ∈ V2, B(W,Φ) =∫D∩Bρ
fφ+ gψ.
It is easily checked that the above right-hand side defines a
continuous linearform on V2, and that
∀ (W,Φ) ∈ V1 × V2, |B(W,Φ)| ≤ ||W || ||Φ||.
21
-
Therefore, the theorem will be proved upon showing that B
satisfies the inf-sup condition (for instance the version in [7]),
i.e., that there exists δ > 0such that
infW∈V1,||W ||=1
(sup
Φ∈V2,||Φ||=1B(W,Φ)
)≥ δ. (38)
Let W = (w, v) ∈ V 1 and p, q, d ∈ R. We set
φ = (Akp+ d)w + (Akq − d)v, ψ = pw + qv,
so that φ−Akψ = d(w − v) = 0, and thus (φ, ψ) ∈ V2 is an
admissible testfunction. The integrand in the expression of B(W,φ)
takes the form
e := ∂ru∂r [(Akp+ d)w + (Akq − d)v] + ∂rv∂r(pw + qv)
+ r−2∂θu∂θ [(Akp+ d)w + (Akq − d)v] +A2
r2∂θv∂θ(pw + qv)
= (Akp+ d)ξ21 + (Akq − d+ p)ξ1ξ3 + qξ23+ (Akp+ d)ξ22 + (Akq −
d+A2p)ξ2ξ4 +A2qξ24 ,
where ξ1 = ∂rφ, ξ2 = r−1∂θφ, ξ3 = ∂rψ, ξ4 = r
−1∂θψ. Fixing q = 1, it followsthat e defines a positive
definite quadratic form (pointwise) provided thatthe
polynomials
P1(ξ) = (Akp+ d) + (Ak − d+ p)ξ + ξ2,P2(ξ) = (Akp+ d) + (Ak −
d+A2p)ξ +A2ξ2,
are strictly positive, in other words, provided that{(Ak − d+
p)2 − 4(Akp+ d) < 0,(Ak − d+A2p)2 − 4A2(Akp+ d) < 0. (39)
We regard these expressions as polynomials in p, the roots of
which arerespectively
f±(d) = (d+Ak)± 2√d(1 +Ak),
g±(d) =1
A2
[(d+Ak)± 2A
√d(1 + k/A)
].
We remark that the roots are real if and only if{k < −1/A if
λ < 0,−A < k < 0 if λ > 0,
22
-
i.e., recalling (34, 29), if and only if λ /∈ [λ−, λ+], which is
our hypothesis.
It only remains to show that we can indeed find parameters p, d
for which (39)is satisfied, i.e. that we can find d such that
(f−(d), f+(d)) ∩ (g−(d), g+(d)) 6= ∅ (40)
(and then pick p in the intersection).
To this end, assume first that −A < k < 0, so that d+ :=
−Ak > 0. Wenote that
f+(d+) + f−(d+)
2= 0,
and that
g+(d+) =2
A2
√Ak(k/A− 1) > 0,
g−(d+) =−2A2
√Ak(k/A− 1) < 0,
which yields (40).
If k < −1/A, one can see that d− = A2 + kA < 0 and
that
f+(d−) + f−(d−)
2= 2Ak +A2 < −1 = gp(d−).
On the other hand, since 0 < A < 1 and k < −1/A, we
have−2A
√d(1 + k/A) < −2
√d(1 +Ak),
so that for any d < 0, g−(d) < f−(d), and in particular
g−(d−) <f+(d−)+f−(d−)
2 .It follows that (40) also holds in this case.
5.3 Proof of Proposition 5
We come back to the singular Weyl sequence uε, which we split as
uε =(1− χρ)uε + χρuε. Proposition 6 shows that (1− χρ)uε converges
stronglyto 0. On the other hand, Theorem applied to χρuε shows
that
||∇(χρuε)||L2(Bρ) ≤ C(||f̃ε||H−1(D∩Bρ) + ||g̃ε||H−1(D∩Bρ)
)≤ C
(||zε||H1(Ω) + ||uε||L2(Ω)
)→ 0.
It thus follows that uε converges strongly to 0 in H1(Ω), which
contradicts
the assumption that ||uε||H1(Ω) = 1, so that β /∈ σess(TD).
23
-
AcknowledgementsThe work of Hai Zhang was supported by HK RGC
grant ECS 26301016and startup fund R9355 from HKUST. E. Bonnetier
was partially supportedby the AGIR-HOMONIM grant from Université
Grenoble-Alpes, and by theLabex PERSYVAL-Lab (ANR-11-LABX-0025-01).
This project was initi-ated while E.B. was visiting Hong Kong
University of Science and Technol-ogy, and completed at the
Institute of Mathematics and its Applications atthe University of
Minnesota. The hospitality and support of both institu-tions is
gratefully acknowledged.
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27
1 Introduction2 The Neumann-Poincaré operator and the Poincaré
variational operator3 Corner singularity functions3.1 Regular
corner singularity functions3.2 Singular corner singularity
functions
4 Construction of singular Weyl sequences5 Characterization of
the essential spectrum5.1 Controling the energy of u away from the
corner5.2 Controling the energy of u near the corner5.3 Proof of
Proposition ??